Partiko still "works" to some value of works. I have been commenting and upvoting with it but only catch glitches sometimes as that's all I do with it. -eta only recently lost notifications with it a couple of days ago.
Only glitch I'm really getting on eSteem is when sometimes the feed fails to load but that's easy fixed by the refresh (pulling down). Otherwise I don't really like using it because the slider goes to too many decimal places and I end up on a completely new screen when doing comments (I like being able to see the whole post when commenting on a post, answering comments is slightly easier sometimes but not others because I don't always get the comment I'm replying to shown at the top).
Haven't tried SteemPeak on mobile but I remember using busy on mobile for ages before Partiko and eSteem happened because it was a lot more responsive (last I checked SteemPeak was a lot better with the responsiveness but still off enough in places to cause problems, but that was a fair while ago now).
I don't know what kind of phone you're using but not being able to tap and see notifications sounds like either a phone problem or a notification settings problem rather than an eSteem problem.
If you want to stay logged in make sure your browser isn't set to clear cookies/local data on closing. I don't know if you have autofill passwords or whatever that setting is called is enabled but that would cut out one step if it's still doing it and anyway it's probably a lot safer if you use LastPass, put in that one password when you open your browser and then click the button after it autofills your login forms for you.
Hope you find a combination of solutions that works for you anyway, little problems like that are so insidiously annoying!
As for the actual quote which I should probably comment on as well seeing as I'm here, even though it's been true for so many other things, I'm grumpybum today and thinking I wish that was true as 17 years trying my absolute damnedest on my pet project and I fee like I haven't budged in the slightest XD
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Reminds me of forum days ;D though I did decide to wait til I got up to answer on my big rig as thinking about doing it on my phone made my brain hurt XD
I know a few that still use busy (or at least post on it) and it seems to be still plodding along. I like having notifications on my phone which is why I have the mobile apps but for posting I think you'd be a lot better off posting through busy (mobile-first design has to be good for something I guess) on phone/tablet which is probably what I'm going to end up doing again (even if it's just comments XD).
The problems I've had with esteem are mostly ux and some niggly things with the actual interface as the notifications work great for me (though lately some seem to be doubling up for no apparent reason).
I've made several adjustments and given up way too much over the years to try to speed things up/make things more efficient/etc, my progress can't add up any faster because I'm permanently stuck in an unresolvable catch-22.
Well it is resolvable but not by a reliable means therefore that doesn't count